Well, it looks to me as if the statement "expressions are the same only if x > 0" should read "Abs[x] > 1" -- which is the domain of ArcSec.
Also, the result for "eq" does not seem helpful because the numerator will always be zero.
I have been trying to do something similar -- getting Mathematica to acknowledge that the formula in my (new) textbook matches.
So:
ClearAll[x, h]
df = ArcSec'[x]
dflim = Limit[(ArcSec[x + h] - ArcSec[x])/h, h -> 0]
dftbl = 1/(Abs[x] (x^2 - 1)^(1/2))
df == dflim (yields true)
dftbl == dflim (yields 1/(Sqrt[-1 + x^2]*Abs[x]) == 1/(Sqrt[1 - 1/x^2]*x^2))
simpdftbl = FullSimplify[dftbl, {x > 1 || x < -1}] (yields 1/Sqrt[x^2*(-1 + x^2)])
simpdflim = FullSimplify[dflim, {x > 1 || x < -1}] (yields 1/Sqrt[x^2*(-1 + x^2)])
simpdftbl == simpdflim (yields true)